Note that the density \(\varrho \)  of the flow in Euler equations (REF ) is a constant. A natural extension of Onsager's conjecture is to consider critical regularity of conservation of energy in the density-dependent (nonhomogeneous) Euler system and the compressible Euler equations.
Important progress involving generalized Onsager's conjecture in both equations has been made (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]} and references therein).
Before we state these results, we recall the nonhomogeneous Euler system (REF )
\(\left\lbrace \begin{aligned}&\varrho _t+\nabla \cdot (\varrho v)=0, \\&(\varrho v)_{t} +\mathrm {div}\,(\varrho v\otimes v)+\nabla \pi =0, \\&\mathrm {div}\,v=0,\end{aligned}\right.\) 
